16.5 and 16.6: Surfaces, Area, and Surface Integrals
                                                           E. Kim
                            Notation follows Thomas’ Calculus: Early Transcendentals (12th Edition) as
                            closely as possible
                            Parametric form
                            Suppose r(u,v) = f(u,v)i + g(u,v)j + h(u,v)k with bounds a ≤ u ≤ b and
                            c ≤ v ≤ d defines a surface S.
                               • The region R: the region in uv-space: a ≤ u ≤ b and c ≤ v ≤ d.
                               • Surface area differential: dσ = kr ×r k dudv.
                                                                 u    v
                               • Surface area integral:
                                        ¨ dσ=¨ kru×rvkdudv=ˆ dˆ bkru×rvkdudv.
                                          S        R                  c  a
                               • General integral: To integrate G(x,y,z) over S,
                                    ¨                ¨                       
                                      S G(x,y,z)dσ =   RG f(u,v),g(u,v),h(u,v) kru ×rvk dudv.
                            Implicit form
                            Sisthesetofpoints(x,y,z)suchthatF(x,y,z) = 0forsomefunctionF(x,y,z)
                               • The region R: the projection of the surface S onto the xy-plane (then
                                 p=k). Or,theprojectionofthesurfaceS ontothexz-plane(thenp = j).
                                 Or, the projection of the surface S onto the yz-plane (then p = i).
                               • Surface area differential: dσ = k∇Fk dA. Here, dA = dxdy if p = k.
                                                                |∇F·p|
                               • Surface area integral:
                                                     ¨ dσ=¨ k∇Fk dA.
                                                       S       R |∇F ·p|
                               • General integral: To integrate G(x,y,z) over S,
                                             ¨ G(x,y,z)dσ =¨ G(x,y,z) k∇Fk dA.
                                               S               R          |∇F ·p|
                            Explicit form
                            S is the set of points (x,y,z) such that z = f(x,y). That is, we look at the
                            graph of some function f.
                               • TheregionR: We’llusuallyneedtopickaboundedsubsetofthedomain
                                 space. That’s R.
                                                               p 2       2
                               • Surface area differential: dσ =  fx +fy +1 dxdy
                               • Surface area integral:
                                                 ¨        ¨ q 2       2
                                                     dσ =      fx +fy +1 dxdy.
                                                   S       R
                               • General integral: To integrate G(x,y,z) over S,
                                      ¨ G(x,y,z)dσ =¨ G(x,y,f(x,y))qf 2+f 2+1 dxdy.
                                                                           x    y
                                        S                R